Dynamic interactions in hydrogen-bonded systems

Journal of Molecular Structure 735–736 (2005) 225–234 www.elsevier.com/locate/molstruc

Dynamic interactions in hydrogen-bonded systems Marek J. Wo´jcik* Faculty of Chemistry, Jagiellonian University, 30-060 Krako´w, Ingardena 3, Poland Received 8 September 2004; revised 7 October 2004; accepted 18 October 2004

Dedicated to Professor Hiroaki Takahashi on occasion of his 70th birthday.

Abstract A theoretical model is presented for the X–H/X–D stretching vibrations in weak and moderately strong hydrogen-bonded systems. The model is based on vibronic-type couplings between high and low frequency modes in hydrogen bridges and Davydov interactions. It allows calculation of energy and intensity distributions in the infrared spectra of hydrogen-bonded systems. The effect of deuterium/hydrogen substitution on the spectra and the temperature effect are explained. Comparison between experimental and theoretical spectra is presented for several systems with hydrogen bonds, crystalline, liquid and gaseous. q 2004 Elsevier B.V. All rights reserved. Keywords: Hydrogen bond; Infrared spectra; Vibrational interactions

1. Introduction

2. Model

Vibrational spectra of hydrogen-bonded systems have been a subject of numerous experimental and theoretical studies [1–22]. Hydrogen bonding brings striking changes in infrared spectra of the X–H(D) stretching bands. These bands are shifted to lower frequencies by amount which reflects the strength of hydrogen bond, and their widths and the total intensities increase by an order of magnitude (see Fig. 1). In this review we present a theoretical model for vibrational couplings in weak and moderately strong hydrogen-bonded systems and use it for modelling experimental infrared spectra for a number of systems with hydrogen bonds, crystalline, liquid and gaseous. The model is based on vibronic-type couplings between high and low frequency modes in hydrogen bridges and Davydov interactions [9,11]. It allows calculation of energy and intensity distributions in the infrared spectra of hydrogenbonded systems. The effect of deuterium/hydrogen substitution on the spectra and the temperature effect are explained by this model.

In our model we treat hydrogen bond as a three atom systems composed of two heavy atoms X and Y (these are usually oxygen, nitrogen, chloride, fluoride or sulphur atoms) and hydrogen or deuterium atom in between these two. The vibrational modes for such system are schematically shown in Fig. 2. There are intramolecular modes: X–H(D) stretching, X–H(D) in-plane bending and X–H(D) out-of-plane bending; and intermolecular modes: X/Y stretching and X/Y bending. In our model we assume an adiabatic coupling between the high-frequency X–H(D) stretching and low-frequency hydrogen-bond stretching vibrations in isolated hydrogen bonds [8]. In a system of interacting hydrogen bonds, such as in hydrogen-bonded dimers or in crystals, there is a degeneracy in the excited state of proton or deuteron stretching vibrations and adiabatic approximation breaks down. In such systems we consider resonance (Davydov) interactions between hydrogen bonds. The form of the Hamiltonian describing vibrations in hydrogen bonds depends on a system. As an example we present the model for a crystal of 1-methylthymine [11]. The structure of 1-methylthymine crystal, taken from Ref. [23], is shown in Fig. 3. In a unit cell there are two

* Tel.: C48 12 663 2913; fax: C48 12 634 05 5. E-mail address: [email protected]. 0022-2860/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2004.10.110

226

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

neglecting their interactions with other modes, and we use one-unit-cell approximation. Also in this model we neglect Fermi resonances between the fundamental nXH(D) stretching and the overtone of the dX–H(D) bending vibrations or combination vibrations. The total vibrational hamiltonian H of the considered system can be written as H Z hA C hB C hC C hD C TA C TB C TC C TD C V; (1) where hx (xZA, B, C, D) is the Hamiltonian of the highfrequency vibration of the xth hydrogen bond, Tx is the kinetic energy operator of the low-frequency vibration, and V describes the resonance interactions between different hydrogen bonds in the unit cell. The total vibrational wavefunction of the hydrogenbonded crystal in the first excited state of the high-frequency vibrations has the form Fig. 1. Infrared spectra of imidazole (a) gaseous at 463 K, (b) liquid at 363 K.

hydrogen-bonded dimers in which molecules are linked by four identical hydrogen bonds: A, B, C, D. They are related ^ by symmetry operations: the screw axis C^ 2 , the inversion I, ^ In our model we consider only two and the slide plane s. modes in each hydrogen bond: high-frequency X–H(D) stretching and low-frequency X/Y stretching vibrations,

Je Z apA C bpB C gpC C dpD ;

(2)

where pA ; pB ; pC ; pD denote wavefunctions of the highfrequency vibrations with the exciton localized on successive hydrogen bonds. Introducing vibrational wavefunction (2) into Schro¨dinger equation with the Hamiltonian (1) and averaging over coordinates of the high-frequency proton vibrations we obtain the following four-dimensional effective dimensionless Hamiltonian expressed in the compact form

H eff Z

3 X 1 iZ1

2

 p2i C q2i 1 C bq1 s1 C bq2 r1 C bq3 s1 r1

C V1 r3 C V2 s3 C V3 s3 r3 C r C H 0 ;

(3)

where qi and pi are symmetry coordinates and conjugated momenta of the low-frequency hydrogen bond vibrations, b is a linear distortion parameter describing change of the equilibrium position of the potential of the low-frequency vibration between ground and excited state of the high-frequency X–H(D) stretching vibration, and Vi (iZ 1–3) describe resonance (Davydov) interactions between hydrogen bond A and B, A and C, and A and D, respectively. Due to symmetry there are only three such parameters. r is the vertical excitation energy of the fast mode and H 0 denotes the Hamiltonian of totally symmetric vibration q0 commuting with the rest of the Hamiltonian Heff: H0 Z Fig. 2. Vibrational modes in hydrogen bond.

 1 2 p0 C q20 1 C bq0 1; 2

(4)

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

227

Fig. 3. A view on the structure of 1-methylthymine crystal along the normal to the (102) plane. Reprinted from Fig. 3, ref. [23], with the IUCr copyright permission.

1 is the unit matrix, and si and ri are the four-dimensional Dirac matrices: 2

0

6 61 s1 Z 6 60 4

1 0 0

0 0

7 0 07 7; 0 17 5

20 0 1 0 0 1 6 60 0 0 r1 Z 6 61 0 0 4 0

3

1 0

03 0 7 17 7; 07 5 0

2

1

0

0

6 6 0 K1 0 s3 Z 6 60 0 1 4

0

3

7 0 7 7; 0 7 5

2 0 0 0 K1 3 1 0 0 0 7 6 0 7 60 1 0 7 r3 Z 6 6 0 0 K1 0 7: 5 4 0

0

0

The IR intensities of the transitions from the ground state to the excited state of the hydrogen bond stretching vibrations are given by the formula m jJej0 ij2 exp Ijj0 wjhJgj j~

(5)



KjZU ; kT

(7)

where jgj is the jth wavefunction of the ground vibrational state of the ns vibrations, jej 0 is the j 0 th wavefunction of the excited vibrational state, and m ~ is the dipole moment.

K1

The energy is expressed in units of the low-frequency quanta. Using symmetry operators C^ 2 and I^ we can reduce fourdimensional Hamiltonian Heff (3) into four one-dimensional Hamiltonians:  3  X  1 2 pi Cq2i Cbqi CV1 I^ CV2 C^ 2 CV3 I^C^ 2 Cr; 2 iZ0

 3 X 1 2  2 CK pi Cqi Cbqi KV1 I^ CV2 C^ 2 KV3 I^C^ 2 Cr; h Z 2 iZ0  3  X  1 2 KC pi Cq2i Cbqi CV1 I^ KV2 C^ 2 KV3 I^C^ 2 Cr; h Z 2 iZ0  3  X  1 2 2 KK pi Cqi Cbqi KV1 I^ KV2 C^ 2 CV3 I^C^ 2 Cr: h Z 2 iZ0

hCC Z

(6) The Hamiltonians hij have give solutions of different symmetry which allows us to calculate polarized spectra of the crystal.

Fig. 4. Comparison between the theoretical and experimental spectra for (a) the 1-methylthymine crystal and (b) the deuterated crystal.

228

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

3. Theoretical modelling of experimental spectra 3.1. 1-Methylthymine crystal

Fig. 5. Comparison between the theoretical and experimental spectra polarised along the b axis of the 1-methylthymine crystal. Solid line: electric vector of the incident radiation parallel to the b axis; dashed line: electric vector of the incident radiation perpendicular to the b axis.

We can also derive expressions for integral properties of the spectra, such as center of gravity u0 and the theoretical half-width Du1/2 u0 Z r K V1 C ðV2 K V3 Þcos 4

Du1=2 Z ½2b2 coth ðZu=2kTÞ C ðV2 K V3 Þsin2 41=2

(8)

where u is the angular frequency of the low-frequency mode, and f is the angle between the dipole transition moments in hydrogen bonds in two different dimers.

Fig. 4 presents the comparison between experimental and calculated spectra for 1-methylthymine crystal and for the deuterated crystal. Calculated spectra are shown as d functions representing calculated transition energies and relative intensities. The calculations have been based on the model described in the Section 2 in its simplest version. In these calculations only the interaction between the nearest hydrogen bonds V1 has been taken into account, i.e. interactions between hydrogen bonds in different dimers have been neglected. Despite this approximation main characteristics of the spectra with regard to the energy and intensity distributions of the fine structure and the width of the spectra are in good agreement with the experimental measurements. According to the theoretical model [9,11] the only parameter which changes after deuteration is the distortion pffiffiffi parameter b which decreases by the ratio 1/ 2. The resonance parameters Vi and the energy quantum of the low-frequency vibrations Zu do not change. Substantially different structure and width of the infrared spectrum of the deuterated 1-methylthymine crystal is correctly reproduced by our calculation, in agreement with these assumptions. Fig. 5 presents polarised experimental spectra of 1-methylthymine crystal parallel and perpendicular to the b axis of the crystal. To calculate polarised spectra we have considered all interactions within unit cell and used Hamiltonians (6). The theoretical parallel-polarised spectra,

Fig. 6. Structures of oxalic acid crystals.

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

229

Fig. 9. Comparison between the theoretical and experimental spectra for (a) oxalic acid dihydrate and (b) deuterated oxalic acid dihydrate crystals.

3.2. Oxalic acid crystals Fig. 7. Comparison between the theoretical and experimental spectra for (a) a-oxalic acid and (b) deuterated a-oxalic acid crystals.

calculated using our model, are shown as d functions in Fig. 5. The main features of the experimental spectrum are well reproduced by model calculations. The details of calculations performed for 1-methylthymine crystal have been presented in Ref. [11].

Fig. 8. Comparison between the theoretical and experimental spectra for (a) b-oxalic acid and (b) deuterated b-oxalic acid crystals.

Oxalic acid crystallizes in three different forms: two anhydrous a and b, and as a dihydrate. Schematic structures of these forms are shown in Fig. 6. Hydrogen bonds in the aoxalic acid form open chain structures separated from one another by the O–CaO groups. In the b-oxalic acid there are cyclic structures with two parallel hydrogen bonds similar to those found in carboxylic acids. The crystalline structure of the dihydrate is similar to the b form but with two molecules of water included, thus forming cyclic structure with four non-equivalent hydrogen bonds. Increasing complexity of the systems of hydrogen bonds is reflected in substantially different infrared bandshapes and fine structures of spectra for O–H(D) stretching vibrations in three crystals. These experimental facts allowed us to devise models for studying three different systems of hydrogen bonds formed by the same molecule in three different crystals. In modelling spectrum of the a-oxalic acid we assumed that hydrogen bonds in this crystal are practically isolated. Theoretical spectra are compared with the experimental ones for OH and OD crystals in Fig. 7. They are composed of the regular Franck-Condon-type progressions in the lowfrequency O/O mode. In the deuterated crystal the pffiffiffi distortion parameter b decreases by 2 and the corresponding progression is shorter. Comparison between theoretical and experimental spectra for the b-oxalic acid and its deuterated analogue is presented in Fig. 8. In the b-oxalic acid two hydrogen bonds in the cyclic structure are coupled. Two high-frequency O–H(D) oscillators coupled in the excited state by the resonance interaction split the n OH(D) band in two

230

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

Fig. 10. Comparison between the theoretical and experimental spectrum of polycrystalline salicylic acid.

Fig. 11. Comparison between the experimental (solid line) and theoretical (d functions) ns spectra for aspirin-H at 300 K.

Fig. 12. Comparison between the experimental (solid line) and theoretical (d functions) ns spectra for aspirin-H at 77 K.

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

231

Fig. 13. Comparison between the experimental (solid line) and theoretical (d functions and broken line) ns spectra for salicylaldehyde OH CHO.

components: one allowed with the parallel orientation of the transition dipole moments - corresponding to higher frequency, and the second forbidden with antiparallel orientation of the transition dipole moments - at lower frequency. The non-adiabatic coupling with low-frequency O/O vibration breaks the selection rules and yields an irregular vibrational fine structure preserving however the bigger intensity on the “allowed” high-frequency side of the spectrum. Fig. 9 presents spectra of the oxalic acid dihydrate. The experimental spectrum shows a prominent broad band in the region 1600–3000 cmK1 for non-deuterated crystal, which becomes narrower after deuteration. In the dihydrate strong coupling between two hydrogen bonds sharing the same oxygen atom in each moiety gives collinear orientation of

the corresponding transition dipole moments. This results in a lower frequency component of the O–H(D) transitions allowed and a higher frequency forbidden. The nonadiabatic coupling with the O/O movement relaxes the selection rules and yields an irregular fine structure preserving however the bigger intensity on the lowerfrequency side of the spectrum. The details of these studies have been presented in Ref. [24]. 3.3. Salicylic acid and its derivatives Another example where we can study infrared spectra of hydrogen bonds in the same family of molecules is salicylic acid, acetylsalicylic acid (aspirin) and salicylaldehyde. In crystals of salicylic acid there are both inter- and

Fig. 14. Comparison between the experimental (solid line) and theoretical (d functions and broken line) ns spectra for salicylaldehyde OH CDO.

232

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

Fig. 15. Comparison between the experimental (solid line) and theoretical (d functions and broken line) ns spectra for salicylaldehyde OD CDO.

Fig. 16. Comparison between the experimental (solid line) and theoretical (dotted line) ns spectra for (a) tropolone-H and (b) tropolone-D.

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

intramolecular hydrogen bonds. Two intermolecular hydrogen bonds in salicylic acid form cyclic structures typical to carboxylic acid dimers. There is also intramolecular strongly distorted hydrogen bond between hydroxyl group in orto position and oxygen of carbonyl group. Experimental and calculated spectra of polycrystalline salicylic acid are presented in Fig. 10. The broad band with fine structure, extending over 1400 cmK1 in hydrogen-bonded crystal, is primarily due to interactions in intermolecular hydrogen bonds. The narrow band, centred at 3243 cmK1, is due to intramolecular bonds. In general, spectra due to intramolecular hydrogen-bonds are more narrow and show less structure than spectra of hydrogen-bonded systems with intermolecular hydrogen bonds, which can be explained by the fact that coupling between high- and low-frequency modes in strongly bent intramolecular hydrogen bonds is smaller. In aspirin (acetylsalicylic acid) there are only intermolecular hydrogen bonds, and the molecules form dimeric cyclic structures similar to those in salicylic acid. The infrared spectra of aspirin and salicylic acid in the region of the O–H stretching vibrations are similar, excluding peaks due to intramolecular hydrogen bonds. The model we used to calculate the spectra included also Fermi resonance between the O–H(D) stretching and the overtone of the O–H bending vibrations, and quadratic distortions of the potential energy for the low-frequency vibrations in the excited state of the O–H stretching vibration. We considered coupling of the O–H(D) stretching vibrations with two low-frequency modes 120 and 323 cmK1. The results of our calculations are presented in Figs. 11 and 12 for two different temperatures 300 and 77 K. The experimental spectra and the temperature effect are reproduced reasonably well.

Fig. 17. Comparison between the experimental (solid line) and theoretical (dashed and dotted line) spectrum of the (CH3)2O/HCl complex at 226 K. Vertical lines show vibrational transitions. Quantum numbers shown relate to the low-frequency hydrogen-bond stretching and bending vibrations.

233

Fig. 18. Comparison between the experimental (solid line) and theoretical (dashed and dotted line) spectrum of the (CH3)2O/HCl complex at 360 K.

A family-related compound with only intramolecular hydrogen bonds is liquid salicylaldehyde. Its experimental spectrum in the O–H stretching region, shown in Fig. 13, exhibits broad band in the range 3000–3600 cmK1 and two narrow but intense peaks at 2751 and 2847 cmK1. They are due to C–H stretching vibrations in the aldehyde group coupled with the O–H stretching. They disappear in the CD derivative of salicylaldehyde (see Fig. 13). We recorded and calculated ns stretching bands of three isotopic species: salicylaldehyde OH CHO, OH CDO, and OD CDO (OD or CDO means deuteration of the hydroxyl or aldehyde group), taking into account in our model resonance interactions between the O–H(D) and C–H(D) stretching vibrations, and quadratic distortions of the potential energy. The results for different isotopic species are presented in Figs. 13–15. Theoretical spectra are represented by both d functions and banshapes calculated with the uniform Gaussian half-widths. The results show that spectra of the liquid salicylaldehyde and its deuterated analogues can be successfully simulated by our model. The details of work on salicylic acid, aspirin and salicylaldehyde are presented in Refs. [25–27].

Fig. 19. Comparison between the experimental (solid line) and theoretical (dashed and dotted line) spectrum of the (CH3)2O/DCl complex at 243 K.

234

M.J. Wo´jcik / Journal of Molecular Structure 735–736 (2005) 225–234

3.4. Tropolone crystal In tropolone molecule there is an intramolecular hydrogen bond between neighbouring OH(D) and C]O groups. In the crystal structure there are additional hydrogen bonds between different molecules, thus OH(D) is engaged both in intra- and intermolecular bonding. The infrared spectra of the tropolone crystals have been calculated assuming coupling of the O–H(D) stretching vibrations with two intra- and intermolecular O/O stretching vibrations. The experimental frequencies of these vibrations have been recorded at 346 and 78 cmK1. The calculations of spectra were done separately for the two modes coupled to the O–H(D) stretches and the calculated bands were convoluted. We present the comparison between the experimental and theoretical spectra for tropolone-H and tropolone-D in Fig. 16. The main broad peak at ca. 3200 cmK1 in non-deuterated crystals is due to the coupling of the O–H stretching with intermolecular mode 78 cmK1, and the peak at 3520 cmK1 is a sideband associated with the low-frequency intramolecular mode 346 cmK1. The calculated spectra reproduce the experimentally observed asymmetry of the ns bands in which most of the combination transitions are located at the low-frequency side of the O–H(D) bands. The details of this work are presented in Ref. [28]. 3.5. Gaseous complex (CH3)2O/HCl Infrared spectra of the dimethyl ether–hydrochloride complex have been studied at different temperatures, and for the deuterated complex. They exhibit a quasi-regular progressions of broad sub-bands of varying intensity ratios (Figs. 17–19). To model spectra we assumed the coupling of the Cl–H(D) stretching vibration with the O/Cl stretching and an intermolecular bending vibration. The potential of the O/Cl stretching vibration was described by a Morse curve. We calculated also rotational transitions in spectra of the gaseous complex by using rotational constants in the ground and excited Cl–H(D) states. Temperature changes in the spectra are the result of changing populations of the initial vibrational states, according to Eq. (7). The calculated spectra are compared with the experimental ones for the (CH3)2/ HCl complex in Figs. 17 and 18, and for the (CH3)2/DCl complex in Fig. 19 [29]. Our model reproduces correctly the temperature dependence of the spectra and the isotopic effect.

4. Conclusions Presented theoretical model of vibrational couplings in hydrogen-bonded systems, taking into account an adiabatic coupling between two stretching vibrations in each hydrogen bond, and Davydov interactions between different hydrogen bonds, allows to reproduce quantitatively main features of experimental spectra for a number of hydrogenbonded systems, in the crystalline, liquid and gaseous states.

The effects of deuteration and temperature are quantitatively reproduced by this model. Further improvements, which require taking into account Fermi resonances, anharmonic potentials and crystal effects, are planned in future.

Acknowledgements Figs. 7–9 are reprinted from Ref. [24], Fig. 10 from Ref. [25], and Figs. 11 and 12 from Ref. [26], with the permission of Elsevier. Figs. 4 and 5 are reprinted from Ref. [11], Figs. 13–15 from Ref. [27], Fig. 16 form Ref. [28], and Figs. 17–19 from Ref. [29], with the permission of John Wiley & Sons. Fig. 1 is reprinted from Ref. [30], with permission of Socite´ Franc¸aise de Chimie. Technical assistance of Mr Łukasz Boda is kindly acknowledged.

References [1] D. Hadzˇi, H.W. Thompson (Eds.), Hydrogen Bonding, Pergamon Press, London, 1959. [2] G.C. Pimentel, A.L. McClellan, The Hydrogen Bond, Freeman, San Francisco, 1960. [3] P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond. Recent Developments in Theory and Experiments vol. I–III (1976). [4] G.A. Jeffrey, W. Sanger, Hydrogen Bonding in Biological Structures, Springer, Berlin, 1990. [5] S. Scheiner (Ed.), Hydrogen Bonding. A Theoretical Perspective, Oxford University Press, Oxford, 1997. [6] G.A. Jeffrey, Introduction to Hydrogen Bonding, Oxford University Press, Oxford, 1997. [7] D. Hadzˇi (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, Chichester, 1997. [8] S. Bratozˇ, D. Hadzˇi, J. Chem. Phys. 27 (1957) 991. [9] Y. Mare´chal, A. Witkowski, J. Chem. Phys. 48 (1968) 3697. [10] A. Witkowski, M. Wo´jcik, Chem. Phys. 1 (1973) 9. [11] M.J. Wo´jcik, Int. J. Quant. Chem. 10 (1976) 747. [12] M.J. Wo´jcik, Mol. Phys. 36 (1978) 1757. [13] G.S. Denisov, G.K. Tokhadze, Dokl. Phys. Chem. 229 (1994) 117. [14] K.G. Tokhadze, G.S. Denisov, M. Wierzejewska, M. Drozd, J. Mol. Struct. 404 (1997) 55. [15] O. Henry-Rousseau, D. Chamma, Chem. Phys. 229 (1998) 37. [16] D. Chamma, O. Henry-Rousseau, Chem. Phys. 229 (1998) 51. [17] M.P. Lisitsa, N.E. Ralko, A.M. Yaremko, Phys. Lett. 40A (1972) 329; M.P. Lisitsa, N.E. Ralko, A.M. Yaremko, Phys. Lett. 48A (1974) 241. [18] H. Ratajczak, A.M. Yaremko, Chem. Phys. Lett. 314 (1999) 122–131. [19] A.M. Yaremko, D.I. Ostrovskii, H. Ratajczak, B. Silvi, J. Mol. Struct. 482–483 (1999) 665. [20] H.T. Flakus, J. Mol. Struct. (Theochem) 285 (1993) 281. [21] H.T. Flakus, A. Bryk, J. Mol. Struct. 372 (1995) 215–229. [22] H.T. Flakus, J. Mol. Struct. 646 (2003) 15. [23] K. Hoogsteen, Acta Cryst. 16 (1963) 28. [24] A. Witkowski, M. Wo´jcik, Chem. Phys. 21 (1977) 385. [25] M.J. Wo´jcik, Chem. Phys. Lett. 83 (1981) 503. [26] M. Boczar, M.J. Wo´jcik, K. Szczeponek, D. Jamro´z, A. Zieba, B. Kawałek, Chem. Phys. 286 (2003) 63. [27] M. Boczar, M.J. Wo´jcik, K. Szczeponek, D. Jamro´z, S. Ikeda, Int. J. Quant. Chem. 90 (2002) 689. [28] M.J. Wo´jcik, M. Boczar, M. Stoma, Int. J. Quant. Chem. 73 (1999) 275. [29] M.J. Wo´jcik, Int. J. Quant. Chem. 29 (1986) 855. [30] A. Novak, J. Chim. Phys. 72 (1975) 981.